Article
On a Vizinglike conjecture for direct product graphs.
Department of Mathematics, PEF, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
Discrete Mathematics (Impact Factor: 0.58). 01/1996; 156:243246. DOI: 10.1016/0012365X(96)000325 Source: DBLP

Article: Dominating direct products of graphs
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ABSTRACT: An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G×H)⩽3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G×H)⩾Γ(G)Γ(H), thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53–79]. Finally, for paireddomination of direct products we prove that γpr(G×H)⩽γpr(G)γpr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.Discrete Mathematics 01/2007; · 0.58 Impact Factor 
Article: Lower bounds for the domination number and the total domination number of direct product graphs
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ABSTRACT: A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: γ(×i=1tKni)≥t+1,t≥3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γt(G×H)=γ(G)+γ(H)−1γt(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made.Discrete Mathematics 12/2010; 310(23):3310–3317. · 0.58 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The problem of determining the domination number of an arbitrary grid graph is known to be NPcomplete, but the complexity of the same problem on complete grid graphs is still unknown. In the present paper we study the same problem on a similar grid graph defined by the cross product of two paths pk and Pn.Discrete Applied Mathematics 01/1999; 94:101139. · 0.72 Impact Factor
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